Dini continuity

Let ${\displaystyle X}$  be a compact subset of a metric space (such as ${\displaystyle \mathbb {R} ^{n}}$ ), and let ${\displaystyle f:X\rightarrow X}$  be a function from ${\displaystyle X}$  into itself. The modulus of continuity of ${\displaystyle f}$  is

${\displaystyle \omega _{f}(t)=\sup _{d(x,y)\leq t}d(f(x),f(y)).}$ 

The function ${\displaystyle f}$  is called Dini-continuous if

${\displaystyle \int _{0}^{1}{\frac {\omega _{f}(t)}{t}}\,dt<\infty .}$ 

An equivalent condition is that, for any ${\displaystyle \theta \in (0,1)}$ ,

${\displaystyle \sum _{i=1}^{\infty }\omega _{f}(\theta ^{i}a)<\infty }$ 

where ${\displaystyle a}$  is the diameter of ${\displaystyle X}$ .

• Dini test — a condition similar to local Dini continuity implies convergence of a Fourier transform.

• Stenflo, Örjan (2001). "A note on a theorem of Karlin". Statistics & Probability Letters. 54 (2): 183–187. doi:10.1016/S0167-7152(01)00045-1.